Computation of transfer function matrices of generalized state-space systems

The discussion of the resolvent matrix (sI-A)(-1) is extended to the generalized resolvent matrix (sE-A)(-1), where E may be singular. The proof of the generalized Leverrier's relation for (sE-A)(-1) is simplified by a closed-form expression of d/ds[det(sE-A)]. For the case of A being singular; the generalized Leverrier's relation can be easily transformed to the generalized Leverrier's algorithm. For the case of both E and A being singular, a new algorithm for (sE-A)(-1) is developed. First, the given regular pencil (sE-A) is converted to a standard pencil by a constant matrix multiplication. Then a change of variable procedure makes the computations convenient. The two matrices problems become one matrix problems after the change of variable step. Hence, the Leverrier's algorithm can be applied. Finally, the rearrangements of scalar and matrix coefficients according to explicit formulae are needed to obtain the desired solution. After (sE-A)(-1) has been computed, the transfer function matrix can be easily obtained. Two numerical examples are used to illustrate our works. This work can be applied to the discrete-time singular system also. The only difference is that the variable is z instead of s.